Tim Seppelt

Two graphs G and H are homomorphism indistinguishable over a class of graphs 𝓕 if for all graphs F ∊ 𝓕 the number of homomorphisms from F to G is equal to the number of homomorphisms from F to H. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes.
For a fixed graph class 𝓕, the decision problem HomInd(𝓕) asks to determine whether two input graphs G and H are homomorphism indistinguishable over 𝓕. The problem HomInd(𝓕) is known to be decidable only for few graph classes F. We show that HomInd(𝓕) admits a randomised polynomial-time algorithm for every graph class 𝓕 of bounded treewidth which is definable in counting monadic second-order logic CMSO2. Thereby, we give the first general algorithm for deciding homomorphism indistinguishability.
This result extends to a version of HomInd where the graph class 𝓕 is specified by a CMSO2-sentence and a bound k on the treewidth, which are given as input. For fixed k, this problem is randomised fixed-parameter tractable. If k is part of the input then it is coNP- and coW[1]-hard. Addressing a problem posed by Berkholz (2012), we show coNP-hardness by establishing that deciding indistinguishability under the k-dimensional Weisfeiler–Leman algorithm is coNP-hard when k is part of the input.

We show that feasibility of the t^th level of the Lasserre semidefinite programming hierarchy for graph isomorphism can be expressed as a homomorphism indistinguishability relation. In other words, we define a class ℒ_t of graphs such that graphs G and H are not distinguished by the t^th level of the Lasserre hierarchy if and only if they admit the same number of homomorphisms from any graph in ℒ_t. By analysing the treewidth of graphs in ℒ_t we prove that the 3t^th level of Sherali-Adams linear programming hierarchy is as strong as the t^th level of Lasserre. Moreover, we show that this is best possible in the sense that 3t cannot be lowered to 3t-1 for any t. The same result holds for the Lasserre hierarchy with non-negativity constraints, which we similarly characterise in terms of homomorphism indistinguishability over a family ℒ_t^+ of graphs. Additionally, we give characterisations of level-t Lasserre with non-negativity constraints in terms of logical equivalence and via a graph colouring algorithm akin to the Weisfeiler-Leman algorithm. This provides a polynomial time algorithm for determining if two given graphs are distinguished by the t^th level of the Lasserre hierarchy with non-negativity constraints.

Two simple undirected graphs are cospectral if their respective adjacency matrices have the same multiset of eigenvalues. Cospectrality yields an equivalence relation on the family of graphs which is provably weaker than isomorphism. In this paper, we study cospectrality in relation to another well-studied relaxation of isomorphism, namely k-dimensional Weisfeiler-Leman (k-WL) indistinguishability. Cospectrality with respect to standard graph matrices such as the adjacency or the Laplacian matrix yields a strictly finer equivalence relation than 2-WL indistinguishability. We show that individualising one vertex plus running 1-WL already subsumes cospectrality with respect to all such graph matrices. Building on this result, we resolve an open problem of Fürer (2010) about spectral invariants.
Looking beyond 2-WL, we devise a hierarchy of graph matrices generalising the adjacency matrix such that k-WL indistinguishability after a fixed number of iterations can be captured as a spectral condition on these matrices. Precisely, we provide a spectral characterisation of k-WL indistinguishability after d iterations, for k,d ∈ ℕ. Our results can be viewed as characterisations of homomorphism indistinguishability over certain graph classes in terms of matrix equations. The study of homomorphism indistinguishability is an emerging field, to which we contribute by extending the algebraic framework of Mančinska and Roberson (2020) and Grohe et al. (2022).

Lovász (1967) showed that two graphs G and H are isomorphic if and only if they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph F, the number of homomorphisms from F to G equals the number of homomorphisms from F to H. Recently, homomorphism indistinguishability over restricted classes of graphs such as bounded treewidth, bounded treedepth and planar graphs, has emerged as a surprisingly powerful framework for capturing diverse equivalence relations on graphs arising from logical equivalence and algebraic equation systems.
In this paper, we provide a unified algebraic framework for such results by examining the linear-algebraic and representation-theoretic structure of tensors counting homomorphisms from labelled graphs. The existence of certain linear transformations between such homomorphism tensor subspaces can be interpreted both as homomorphism indistinguishability over a graph class and as feasibility of an equational system. Following this framework, we obtain characterisations of homomorphism indistinguishability over several natural graph classes, namely trees of bounded degree, graphs of bounded pathwidth (answering a question of Dell et al. (2018)), and graphs of bounded treedepth.